Adsorption and collapse transitions of a linear polymer chain interacting with a surface adsorbed polymer chain (original) (raw)
We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer bbb ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents gamma1,gamma11\gamma_1, \gamma_{11}gamma1,gamma11, and gammas\gamma_sgammas which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method, for 2leble42\le b\le 42leble4, we have obtained specific values for these exponents, for various type of polymer conformations. We discuss their mutual relations and their relations with other critical exponents pertinent to SAWs on the SG fractals.
Journal of Statistical Mechanics-theory and Experiment, 2008
We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three--dimensional Sierpinski Gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modelled by two interacting self-avoiding walks (SAWs), one of them representing a 3D floating polymer, while the other corresponds to a chain adhered to one of the four 2D SG boundaries. We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the inter-chain interactions we propose two models: in the first model (ASAWs) the SAW chains are mutually avoiding, whereas in the second model (CSAWs) chains can cross each other. By applying an exact Renormalization Group (RG) method, we establish the relevant phase diagrams for b=2,3b=2,3b=2,3 and b=4b=4b=4 members of the 3D SG fractal family for the model with avoiding SAWs, and for b=2b=2b=2 and b=3b=3b=3 fractals for the model with crossing SAWs. Also, at the appropriate transition fixed points we calculate the contact critical exponents, associated with the number of contacts between monomers of different chains. Throughout the paper we compare results obtained for the two models and discuss the impact of the topology of the underlying lattices on emerging phase diagrams.
Journal of Physics A-mathematical and General, 2003
We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) Sierpinski gasket (SG) family of fractals. Each member of the 3d SG fractal family has a fractal impenetrable 2d adsorbing surface (which is, in fact, 2d SG fractal) and can be labelled by an integer bbb ($2\le b\le\infty$). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents nu\nunu (associated with the mean squared end-to-end distance of polymers) and phi\phiphi (associated with the number of adsorbed monomers), for a sequence of fractals with 2leble42\le b\le42leble4 (exactly) and 2leble402\le b\le402leble40 (Monte Carlo). We find that both nu\nunu and phi\phiphi monotonically decrease with increasing bbb (that is, with increasing of the container fractal dimension dfd_fdf), and the interesting fact that both functions, nu(b)\nu(b)nu(b) and phi(b)\phi(b)phi(b), cross the estimated Euclidean values. Besides, we establish the phase diagrams, for fractals with b=3b=3b=3 and b=4b=4b=4, which reveal existence of six different phases that merge together at a multi-critical point, whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface.
Journal of Statistical Physics, 1996
We study the problem of polymer adsorption in a good solvent when the container of the polymer-solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer b (2<~b~< or), and it is assumed that one side of each SG fi'actal is an impenetrable adsorbing boundary. We calculate the critical exponents ),j, ?l J, and ),~, which, within the self-avoiding walk model (SAW) of tile polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2 ~< b ~< 8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2 ~<b ~<80, we obtain specific values for these exponents. The obtained results show that all three critical exponents y~, ~'ll, and y.~, in both the bulk phase and crossover region are monotonically increasing functions with b. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limit b ~ or, when the fractal dimension of the SG fractals approaches the Euclidean value 2.
Critical dynamics of a polymer chain in a grafted monolayer
Macromolecules, 1991
We report on the conformational properties and transitions of an ideal polymer chain near a solid surface. The chain is tethered with one of its ends at distance z 0 from an adsorbing surface. The surface is characterized by an adsorption parameter c. The exact expression for the partition function is available. We obtained the distribution of complex zeros of this function. The comparison with the Yang-Lee theory allows the characterization of the phase transitions. A first-order conformational transition from a coil to a ͑adsorbed͒ flower conformation occurs at c*ϭ6z 0 /N. The flower is composed of a strongly stretched stem and a pancake that collects the remaining adsorbed segments. The degree of stretching of the coil or of the stem serves as an order parameter which parametrizes the analytical expressions of the Landau free energy. The phase diagram with one binodal and two spinodal lines is presented. The height of the barriers between metastable and stable states is obtained and the lifetime of metastable states is estimated. A two-state ansatz is used to develop scaling arguments to account for the effects of excluded volume.
Conformational Properties of Polymers Near a Fractal Surface
Physics Procedia, 2012
The conformational properties of flexible polymer macromolecules grafted to an attractive partially penetrable surface with fractal dimension d p c s = 91/49 are studied. Employing computer simulations based on the prunedenriched Rosenbluth chain-growth method, estimates for the surface crossover exponent and adsorption transition temperature are found. Our results quantitatively reveal the slowing down of the adsorption process caused by the fractal self-similar structure of the underlying substrate.
We present a real space renormalization group (RSRG) approach to a model of selfavoiding walks (SAW) with attractive interactions, which is expected to describe the O-point transition of linear polymers. Both in two and in three dimensions the results of our approximations are physically sound and qualitatively consistent with other numerical predictions and theoretical conjectures. In particular the collapsed phase is correctly described in the present approach.
2009
We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension dfd_fdf is equal to 2 for all members of the fractal family enumerated by the odd integer bbb ($3\le b< \infty$). For various values of stiffness parameter sss of the chain, on the PF fractals (for 3leble93\le b\le 93leble9) we calculate exactly the critical exponents nu\nunu (associated with the mean squared end-to-end distances of polymer chain) and gamma\gammagamma (associated with the total number of different polymer chains). In addition, we calculate nu\nunu and gamma\gammagamma through the MCRG approach for bbb up to 201. Our results show that, for each particular bbb, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of sss) display enlarged values of nu\nunu, and diminished values of gamma\gammagamma. On the other hand, for any specific sss, the critical exponent nu\nunu monotonically decreases, whereas the critical exponent gamma\gammagamma monotonically increases, with the scaling parameter bbb. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.